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Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions$f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain$\Omega$ containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ intoa complex Banach space $X$ with $\norm{f}_{H^{\infty}(\Omega,X)} \leq 1$. Let$\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that$\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to$r \in [0,1)$. For $1\leq p,q<\infty$, we denote  \begin{equation*}  R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \norm{x_{0}}^p \phi_{0}(r) +\left(\sum_{n=1}^{\infty} \norm{x_{n}}\phi_{n}(r)\right)^q \leq\phi_{0}(r)\right\}  \end{equation*}  and define the Bohr radius associated with $\phi$ by$$R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X):\norm{f}_{H^{\infty}(\Omega,X)} \leq 1\right\}.$$  In this article, we extensively study the Bohr radius$R_{p,q,\phi}(\Omega,X)$, when $X$ is an arbitrary Banach space and $X$ iscertain Hilbert space. Furthermore, we establish the Bohr inequality for theoperator-valued Ces\'{a}ro operator and Bernardi operator.

Bohr radius for Banach spaces on simply connected domains